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Extension of bounded vector-valued functions. (English) Zbl 1177.46024
This article studies the following problem about the extension of bounded holomorphic or harmonic functions. Let Ω be an open set and let E be a locally complete locally convex space. Find conditions on AΩ, and HE ' to assure that every bounded function f:AE such that uf:A which has an extension in (Ω) for each uH is the restriction to A of a vector valued function F (Ω,E). The authors show that the extension is possible if (1) A is a set of uniqueness for (Ω) and H determines boundedness in E, or if (2) A is a sampling set for (Ω) and H is weak*-dense in E ' . As a corollary, they obtain a vector valued Blaschke theorem which complements a result due to W. Arendt and N. Nikolski [Math. Z. 234, 777–805 (2000; Zbl 0976.46030)]. The proofs of these results are based on deep functional analytic techniques which continue an approach started by the first two authors and the reviewer in [Stud. Math. 183, 225–248 (2007; Zbl 1141.46017)].
MSC:
46E40Spaces of vector- and operator-valued functions
46E10Topological linear spaces of continuous, differentiable or analytic functions
46A32Spaces of linear operators; topological tensor products; approximation properties